1. Field of the Invention
The present invention relates to digital signal processing techniques, noise filters, nuclear magnetic resonance spectroscopy, and nuclear magnetic resonance imaging, and particularly to a method for removing noise from nuclear magnetic resonance signals and images.
2. Description of the Related Art
Nuclear magnetic resonance (NMR) is a potentially useful and effective diagnostic tool in basic research, clinical investigation and disease diagnosis, since it provides both chemical and physiological information regarding the tissue under investigation. NMR is based upon the precession of protons when placed in a magnetic field. The precessional axis lies along the direction of the magnetic field, and the precession occurs with a frequency directly proportional to the strength of the magnetic field, with a proportionality constant unique to each atom, which is called the gyromagnetic ratio.
If an oscillating magnetic field at the precessional frequency is applied in a direction perpendicular to the static field, the protons will precess about the axis of the oscillating field, as well as that of the static field. The oscillating field is generated by a tuned radio-frequency (RF) coil. The magnetic field of the precessing protons induces, in turn, an oscillating voltage in the RF coil, which is detected when the RF field is gated off. This voltage has an exponential decay over time and is generally referred to as the free induction decay (FID).
This voltage is then amplified and demodulated to baseband, similar to a superheterodyne receiver, and digitized using an analog-to-digital converter. The oscillating RF pulse is repeated and the FIDs are added coherently.
In order to factor out the signal's dependence upon the static magnetic field, NMR measurements are often given in a unitless quantity measured in parts per million (ppm), known as the chemical shift, δ, which is the difference between the precession frequency of protons that are part of a particular molecular group and that of protons in a reference compound, divided by the latter; i.e.,
  δ  =                              f          sample                -                  f          reference                            f        reference              ×          10      6        ⁢          ppm      .      This particular application of NMR is generally referred to as high-resolution nuclear magnetic resonance spectroscopy, and is widely used in the pharmaceutical and chemical industries.
NMR has been extended to the in vivo study of human anatomy. This has been made possible through practical methods for exciting signals from limited volumes and for generating spatial maps of this signal. The protons of the water molecules of the patient's tissue are the source of the signal in magnetic resonance imaging (MRI). Signal strength is modified by properties of each proton's microenvironment, such as its mobility and the local homogeneity of the magnetic field. The spatial information needed to form images from magnetic resonance is obtained by placing magnetic field gradient coils on the inside of the magnet. These coils create additional magnetic fields that vary in strength as a linear function of distance along the three spatial axes.
Thus, the resonant frequencies of the water protons within the patient's body are spatially encoded. The contrast in MRI images arises from difference in the number of protons in a given volume and in their relaxation times (the time taken for the magnetization of the sample to return to equilibrium after the RF pulse is gated off), which are related to the molecular environment of the protons.
The signal-to-noise ratio (SNR) of NMR signals and images is typically the limiting factor for higher data interpretation. Signal/image averaging is typically used to increase the SNR, but this method often results in a substantial increase in the acquisition time, which may not be tolerable in many situations, particularly for unstable biological compounds where long replications are not feasible. Further, a longer time is not desirable with regard to patient comfort, as well as avoidance of motion artifacts in medical MRI usage. Thus, any reduction in acquisition time without compromising the SNR is valuable in terms of decreasing the high cost of the NMR machine time.
Typically, NMR signal/image denoising is performed by either denoising algorithms based on singular value decomposition (SVD), or denoising via thresholding in time-frequency (TF) or time-scale (TS) domains. The TS domain is also known as the “wavelet” domain.
SVD based algorithms are commonly found in NMR denoising schemes. One such algorithm, the Cadzow algorithm, is based upon the following principle: a noiseless time-domain signal including Q exponentially decaying sinusoids can be identified by a Hankel matrix of rank Q. The measurement from NMR is often of full rank because it is inevitable that random noises have been added. Thus, the goal of denoising is to retrieve as much information as possible about the noiseless Hankel matrix from the observed matrix. The truncated SVD produces the most optimal low rank approximation.
However, this truncated SVD of the observed matrix is generally no longer a Hankel matrix. One method of retrieving a Hankel low rank approximation is to alternate projections between the manifold of rank-Q matrices and the space of Hankel matrices. Such a method may remove any corrupting noise, measurement distortion or theoretical mismatch present in the given data set.
Using TF and TS analysis in NMR denoising is a relatively new methodology. The procedure exploits the fact that TF and TS transforms map white noise in the signal domain to white noise in the transform domain. Thus, although signal energy becomes more concentrated into fewer coefficients in the transform domain, noise energy does not. This important principle enables the separation of signal from noise as follows: (1) transform the noisy signal into a new domain; (2) retain only the coefficients whose magnitudes are above a certain threshold related to the known or estimated noise standard deviation; and (3) perform the synthesis transform on the retained coefficients to obtain the noise-reduced signal.
The transform is chosen according to its ability to represent the NMR signal/image in a small number of coefficients, which depends on the characteristics of both the NMR signal/image and the transform. For example, if the NMR signal/image spans 75% of the transform coefficients, then 25% of the noise could be removed through proper filtering. The retained coefficients, however, still contain approximately 75% of the noise in addition to the desired signal/image.
One of several common transforms used in NMR denoising is the wavelet transform, which has become a popular tool in MRI denoising applications. The wavelet transform localizes the signal/image in both frequency (wave number) and time (position) simultaneously.
In an article co-authored by the present inventor and published in the IEEE Transactions on Medical Imaging, 20(10) pp. 1018-1025 (October 2001), it is suggested that stable linear time-frequency (SLTFN) transforms (described in further detail below, with particular reference to equation 1) show a more compact NMR signal/image representation than wavelet transforms and, hence, superior NMR denoising results with minimal desired signal/image distortion. With regard to NMR signal/image denoising, the SLTFN transform provides more optimal results than other TF transforms. The SLTF is a linear TF transform. Thus, it is relatively easy to compute the inverse transform. This is in contrast to bilinear distributions where difficulties are encountered in retrieving the signal from the TF-domain.
Further, there is no cross-term interference in the SLTF transform such as that encountered in the bilinear distributions. In addition, SLTFN is a critically-sampled transform, which exceeds over-sampled transforms in ease and effectiveness of signal retrieval from the transformed domain after denoising. Compared to other critically-sampled TF transforms, SLTF has two major advantages: high numerical stability and excellent localization of the biorthogonal function. In addition, a fast algorithm to calculate the SLTF and inverse transforms (described in an article authored by the present inventor appearing in the Proceedings of the 11th IEEE Workshop on Statistical Signal Processing, August 2001, pp. 317-320) may be utilized, which is linearly proportional to the signal/image size.
Several thresholding techniques have been developed, such as “hard thresholding” where one retains only the coefficients whose magnitudes are above a threshold proportional to the known or estimated noise power, “soft thresholding”, and “global thresholding.”
None of the above inventions, taken either singly or in combination, is seen to describe the instant invention as claimed. Thus, a method for removing noise from nuclear magnetic resonance signals and images solving the aforementioned problems is desired.